Numerical Solution of the Quasilinear Poisson Equation in a Nonuniform Triangle Mesh

Winslow, A.M.

doi:10.1006/jcph.1997.5698

Cited by 225 publications

(217 citation statements)

References 6 publications

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“…One of the simplest choices of monitor functions is G 1 = G 2 = w I , where I is the identity matrix and w > 0 is a weight function, for example w = 1 + u 2 x + u 2 y . In this case, we obtain Winslow's variable diffusion method [47]:…”

Section: Classical Variational Mesh Generationmentioning

confidence: 99%

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Ceniceros¹,

Hou²

2001

Journal of Computational Physics

148

156

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We develop an efficient dynamically adaptive mesh generator for time-dependent problems in two or more dimensions. The mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic PDEs for the mesh map. When coupled to a physical problem, the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior. The overall mesh strategy is simple to implement, avoids interpolation, and can be easily incorporated into a broad range of applications. The efficacy of the mesh is first demonstrated by two examples of blowing-up solutions to the 2-D semilinear heat equation. These examples show that the mesh can follow with high adaptivity a finite-time singularity process. The focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2-D Boussinesq fluid, a challenging problem. The moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically. The numerical results show the fast collapse to small scales and an exponential vorticity growth.

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Section: Classical Variational Mesh Generationmentioning

confidence: 99%

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Ceniceros¹,

Hou²

2001

Journal of Computational Physics

148

156

View full text Add to dashboard Cite

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“…By application of the Winslow method [47] a second order differential equation is used to define the rate of change of the spatial frame of reference with respect to the material frame of reference (Equation (12)), from which the ALE gradient ψ is subsequently used to specify motion (Equation (13)):…”

Section: Arbitrary Lagrangian-eulerian Formulationmentioning

confidence: 99%

An Arbitrary Lagrangian–Eulerian Formulation for Modelling Cavitation in the Elastohydrodynamic Lubrication of Line Contacts

Boer

Dowson

2018

Lubricants

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Abstract:In this article an arbitrary Lagrangian-Eulerian (ALE) formulation for modelling cavitation in elastohydrodynamic lubrication (EHL) is derived and applied to line contact geometry. The method is developed in order to locate the position of cavitation onset along the length of the contacting region which gives the transition from liquid to vapour in the fluid. The ALE is implemented by introducing a spatial frame of reference in which the solution is required and a material frame of reference in which the governing equations are solved. The spatial frame is moved from the material frame according to the error in the Neumann pressure gradient constraint required at the cavitation location when Dirichlet constraints are imposed for pressure in the liquid phase. Results are calculated under both steady-state and transient operating conditions using a multigrid solver. The solutions obtained are compared to established literature and conventional approaches to modelling cavitation which show that the ALE formulation is an alternative, straightforward and accurate means of implementing such conditions in EHL. This is achieved without the penalties associated with the numerical modelling of Heaviside functions or free boundaries.

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“…The Winslow Functional In this approach, the goal is to construct a parametrization as conformal as possible [33,11,22]. The parametrization X is conformal if and only if the Jacobian J is the product of a scaling and a rotation, or, equivalently, if the first fundamental form g is diagonal with identical diagonal elements.…”

Section: Geometric Measuresmentioning

confidence: 99%

Planar Parametrization in Isogeometric Analysis

Gravesen

Evgrafov

Nguyen

et al. 2014

Mathematical Methods for Curves and Surfaces

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Abstract. Before isogeometric analysis can be applied to solving a partial differential equation posed over some physical domain, one needs to construct a valid parametrization of the geometry. The accuracy of the analysis is affected by the quality of the parametrization. The challenge of computing and maintaining a valid geometry parametrization is particularly relevant in applications of isogemetric analysis to shape optimization, where the geometry varies from one optimization iteration to another. We propose a general framework for handling the geometry parametrization in isogeometric analysis and shape optimization. It utilizes an expensive non-linear method for constructing/updating a high quality reference parametrization, and an inexpensive linear method for maintaining the parametrization in the vicinity of the reference one. We describe several linear and non-linear parametrization methods, which are suitable for our framework. The non-linear methods we consider are based on solving a constrained optimization problem numerically, and are divided into two classes, geometry-oriented methods and analysisoriented methods. Their performance is illustrated through a few numerical examples.

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Numerical Solution of the Quasilinear Poisson Equation in a Nonuniform Triangle Mesh

Cited by 225 publications

References 6 publications

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

An Arbitrary Lagrangian–Eulerian Formulation for Modelling Cavitation in the Elastohydrodynamic Lubrication of Line Contacts

Planar Parametrization in Isogeometric Analysis

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